3 Topology of the complex plane
In this section we explain that most of the facts about limits, series, and continuity carry over from real analysis essentially without change.
The modulus of complex numbers defines a distance \(d(z,w)=|z-w|\) on the plane (this is the usual Euclidean distance), which determines the following standard terminology for metric spaces.
Definition 3.1 The open disc of radius \(0\leqslant r\leqslant+\iy\) centered at \(z_0\in\C\) is
\[D_r(z_0)=\{z\in\C \mid |z-z_0|<r\}.\]
The closed disc \(\ol{D}_r(z_0)\) is the set of all \(z\in\C\) with \(|z-z_0|\leqslant r.\)
Definition 3.2 A subset \(O\subset\C\) is open if for every point \(z_0\in O\) there exists \(r>0\) such that \(D_r(z_0)\subset O\) (see Figure 3.1). A subset \(C\subset \C\) is called closed if the complement \(O=\C\setminus C\) is an open subset.
A subset \(B\subset\C\) is bounded if \(B\subset D_r(0)\) for some \(0<r<\iy.\)
Remark 3.1. The above notion of open set determines a topology on \(\C.\)
Definition 3.3 A sequence of complex numbers \((z_n)_{n\in\N}\) has the limit \(\ze\in\C\) (or is convergent to \(\ze\)), written \(\lim_{n\to\iy}z_n=\ze,\) if \[\forall\ep>0\;\exists n_0\in\N\;\forall n\geqslant n_0: |z_n-\ze|<\ep.\]
Equivalently, \(\lim_{n\to\iy}|z_n-\ze|=0.\) We call \((z_n)_{n\in\N}\) a Cauchy sequence if \[\forall\ep>0\;\exists n_0\in\N\;\forall n,m\geqslant n_0: |z_n - z_m|<\ep.\]
The same argument as in real analysis shows that the limit \(w\) is unique and that every convergent sequence is a Cauchy sequence. The converse is also true by the completeness of real numbers.
Proposition 3.1 For a sequence \((z_n)_{n\in\N}\) in \(\C,\) the following are equivalent:
- There exists \(\ze\in\C\) such that \(\ze=\lim_{n\to\iy}z_n.\)
- \((z_n)_{n\in\N}\) is a Cauchy sequence.
Proof.
To prove (b)\(\implies\)(a) write \(z_n=x_n+iy_n\) and notice that \[|z_n-z_m|=\sqrt{(x_n-x_m)^2+(y_n-y_m)^2}\geqslant|x_n-x_m|,\|y_n-y_m|\]
implies that both \((x_n)_{n\in\N}\) and \((y_n)_{n\in\N}\) are Cauchy sequences in \(\R.\) By the completeness of \(\R,\) these sequences have limits \(\chi,\upsilon \in \R.\) Set \(\ze=\chi+i\upsilon\) and pick \(n_0\in\N\) such that \(|x_n-\chi|<\frac{\ep}{\sqrt{2}}\) and \(|y_n-\upsilon|<\frac{\ep}{\sqrt{2}}\) for all \(n\geqslant n_0.\) Then \[|z_n-\ze| = \sqrt{(x_n-\chi)^2+(y_n-\upsilon)^2}<\sqrt{\ep^2/2+\ep^2/2}=\ep\]
for all \(n\geqslant n_0.\) The converse, (a)\(\implies\)(b), is left as an exercise.
The advantage of Cauchy sequences is that one does not need to know the value of the limit in advance.
Definition 3.4 A series of complex numbers \((z_k)_{k\in\N}\) converges to the limit \(\ze,\) written \(\ze=\sum_{k=0}^\iy z_k,\) if the sequence of partials sums \(\bigl(w_n=\sum_{k=0}^n z_k\bigr)_{n\in\N}\) converges to \(\ze.\) We call a series absolutely convergent if the series \(\sum_{k=0}^\iy |z_k|\) is convergent.
As in real analysis, the Cauchy criterion implies that every absolutely convergent series is convergent. Absolutely convergent series may be rearranged and orders of summation may be exchanged.
Although \(\iy\notin\C,\) it will be convenient to define \(\lim_{n\to\iy}z_n=\iy\) to mean that the sequence \((z_n)_{n\in\N}\) eventually leaves every disk. Symbolically, \[\forall r>0\;\exists n_0\in\N\;\forall n\geqslant n_0: |z_n|>r.\]
We call \(\C\cup\{\iy\}\) the extended complex plane.
Definition 3.5 A point \(\ze\in\C\cup\{\iy\}\) is in the closure of \(D\subset \C\) if there exists a sequence \((z_n)_{n\in\N}\) with \(z_n\in D\) and \(\lim_{n\to\iy} z_n=\ze.\)
Let \(f\colon D\to\C\) be a complex function and let \(\ze\in\C\cup\{\iy\}\) be in the closure of \(D.\) The function \(f(z)\) has the limit \(w\in\C\cup\{\iy\}\) as \(z\to\ze,\) written \(\lim_{z\to\ze}f(z)=w,\) if for every sequence \((z_n)_{n\in\N}\) with \(z_n\in D\) and \(\lim_{n\to\iy} z_n=\ze\) we have \(\lim_{n\to\iy}f(z_n)=w.\) An equivalent \(\varepsilon\)-\(\de\)-definition is \[\forall\varepsilon>0\;\exists\de>0: 0<|z-\ze|<\de, z\in D\implies |f(z)-w|<\varepsilon.\]
A complex function \(f\colon D\to\C\) is continuous at \(\ze\in D\) if \(\lim_{z\to\ze}f(z)=f(\ze).\) We call \(f\) continuous on \(D\) if \(f\) is continuous at every \(\ze\in D.\)
Questions for further discussion
- Explain the difference between the notions ‘limit of a function’, ‘limit of a sequence’, and ‘limit of a series’.
3.1 Exercises
Exercise 3.1
Recall the ratio and root test for series of real numbers.
Exercise 3.2
For which \(z\in\C\) do the following limits exist? \[\lim_{n\to\iy}n^{1/n}z, \lim_{n\to\iy}z^n, \lim_{n\to\iy}\frac{z^n}{n}, \lim_{n\to\iy}\frac{z^n}{n!}, \lim_{n\to\iy}\frac{z^n}{n^n}, \lim_{n\to\iy}n!z.\]
Exercise 3.3
Show that every convergent sequence \((z_n)_{n\in\N}\) of complex numbers is bounded.
Exercise 3.4
Let \(\sum_{k=0}^\iy z_k\) be a convergent series of complex numbers. Show that \(\lim_{k\to\iy} z_k=0.\)
Exercise 3.5
For which \(z\in\C\) do the following series converge? \[\sum_{k=0}^\iy kz^k,\quad \sum_{k=0}^\iy(kz)^k\]
Exercise 3.6
Let \(f\colon\C\to\C\) be a complex function. Show that \[\lim_{z\to\iy}f(z)=w\iff\lim_{z\to0}f\left(\frac{1}{z}\right)=w.\]
Exercise 3.7
The Riemann sphere is \(\mathbb{S}=\{(a,b,c)\in\R^3\mid a^2+b^2+c^2=1\}.\) Show that the stereographic projection \[F\colon\mathbb{S}\longra\C\cup\{\iy\}, F(a,b,c)= \begin{cases} \frac{a+ib}{1-c} & \text{ if }c\neq 1,\\ \iy & \text{ if }c=1. \end{cases}\]
is a bijection between the Riemann sphere and the extended complex plane. Find a formula for the inverse function.
